We write \({u}^\Delta (x,t) = \sum _{|{k}|_\infty \le N} \widehat{{u}}^\Delta _{{k}}(t) e^{i{k}\cdot {x}}\) , where now and in the following we shall consistently denote \(\Delta = 1/N\) , and we denote \(|{k}|_\infty {:=}\max _{i=1,\dots ,d} |k_i|\) . We consider the following spectral viscosity approximation [1]}, [2]} of the incompressible Euler equations,
\( \left\lbrace \begin{aligned}\partial _t {u}^\Delta +{\mathcal {P}}_N({u}^\Delta \cdot \nabla {u}^\Delta )+ \nabla p^\Delta &=-\epsilon _N |\nabla |^{2\sigma } (Q_N \ast {u}^\Delta ),\\{\mathrm {div}}({u}^\Delta )&=0,\\{u}^\Delta |_{t=0}&={\mathcal {P}}_N \overline{u}.\end{aligned}\right.\) 
